Pell Mell

Submitted by Billy from Montreal, QC, 3/4/2000. Original answer by Carl Johan Ragnarsson; this article by Allen Stenger.

Show that for any the number can be written as the difference of the square roots of two consecutive integers.

Hint 1

Work out the first few powers and verify the result for them. Do you see any patterns?

Hint 2

Here are the first 5 powers:

first 5 powers of sqrt(2) - 1

If we turn this around we get a nicer pattern:

first 5 powers of 1 - sqrt(2)

Based on these examples, it appears that the result is correct, and in fact the first square root is the square root of a perfect square. Do you see any relationships between the numbers for different powers?

Hint 3

Let's write

From the table it appears that each A value is about twice the previous value, and in fact it appears that the difference is

Also it appears that

This table swarms with recursions! See if you can prove some of them.

Hint 4

Since the thing we are interested in (a power) has a recursive definition, we can use that to prove the recursions:

proof of recursions

and therefore

Now see if you can prove the difference in the squares is 1 or -1.

The Rest of the Solution

Again we proceed by recursion: If it is true for n, then

proof it is the difference of consecutive squares

Is This a Surprising Result?

What special property does have, that its powers are always the difference of the squares roots of two consecutive integers? It's easy to see that any power of has the same property; are there other kinds of numbers with this property? Yes! Here are some even more unlikely numbers with this property:

more magical numbers

What's going on here? We can get a better understanding by approaching the problem from the other end. In our proof we dealt with an equation of the form

(for N=2). We can factor this as

and raise each term to a power

When you multiply these powers out, you will see that the terms have the form

that is,

In other words, any solution of this equation leads to additional solutions, by raising a certain expression to a power, and all powers of this expression are the difference of the square roots of two consecutive integers.

The equation

is called Pell's equation, and there is a great deal of Pell lore in Number Theory. We've seen just a little bit of this lore here; there is much more in connection with recursions, rational approximations to square roots, and continued fractions. See Shanks's book for more.

References

Edmund Landau, Elementary Number Theory, AMS Chelsea, 1966, pp. 76-84. Many books on number theory cover Pell's equation; this book gives an especially clear and simple development.
Daniel Shanks, Solved and Unsolved Problems in Number Theory, AMS Chelsea, 1993. Pell's equation is spread all through Chapter III, but see especially Theorem 77 and the surrounding pages.
Click here to view the original problem submitted by Billy.